# 121

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## Number

$121$ (**one hundred and twenty-one**) is:

- $11^2$

- The $1$st of the two square numbers which is the sum of consecutive powers of a positive integer:
- $121 = 3^0 + 3^1 + 3^2 + 3^3 + 3^4$

- The $2$nd power of $11$ after $(1)$, $11$:
- $121 = 11^2$

- The $2$nd square number after $25$ of the form $n! + 1$:
- $121 = 5! + 1 = 11^2$

- where $!$ denotes the factorial function.

- The $2$nd Fermat pseudoprime to base $3$ after $91$:
- $3^{121} \equiv 3 \pmod {121}$

- The $2$nd (with $4$) of the $2$ square numbers which are $4$ less than a cube:
- $121 = 11^2 = 5^3 - 4$

- The $2$nd Wonderful Demlo number after $1$

- The $4$th square number after $1$, $4$, $36$ to be the divisor sum value of some (strictly) positive integer:
- $121 = \map {\sigma_1} {81}$

- The $6$th integer after $0$, $1$, $2$, $4$, $8$ which is palindromic in both decimal and ternary:
- $121_{10} = 11 \, 111_3$

- The $7$th Smith number after $4$, $22$, $27$, $58$, $85$, $94$:
- $1 + 2 + 1 = 1 + 1 + 1 + 1 = 4$

- The $9$th positive integer which cannot be expressed as the sum of a square and a prime:
- $1$, $10$, $25$, $34$, $58$, $64$, $85$, $91$, $121$, $\ldots$

- The $9$th palindromic integer after $0$, $1$, $2$, $3$, $11$, $22$, $101$, $111$ whose square is also palindromic integer
- $121^2 = 14 \, 641$

- The $10$th square after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$ which has no more than $2$ distinct digits and does not end in $0$:
- $121 = 11^2$

- The $11$th square number after $1$, $4$, $9$, $16$, $25$, $36$, $49$, $64$, $81$, $100$:
- $121 = 11 \times 11$

- The $16$th powerful number after $1$, $4$, $8$, $9$, $16$, $25$, $27$, $32$, $36$, $49$, $64$, $72$, $81$, $100$, $108$

- The $39$th semiprime:
- $121 = 11 \times 11$

## Also see

- Brocard's Problem
- Squares which are 4 Less than Cubes
- Square Numbers which are Sum of Consecutive Powers
- Integer Greater than 121 is Sum of Distinct Primes of form 4 n + 1

*Previous ... Next*: Wonderful Demlo Number*Previous ... Next*: Palindromes in Base 10 and Base 3*Previous ... Next*: Sequence of Powers of 11*Previous ... Next*: Square Numbers which are Divisor Sum values*Previous ... Next*: Squares with No More than 2 Distinct Digits*Previous ... Next*: Numbers not Sum of Square and Prime*Previous ... Next*: Fermat Pseudoprime to Base 3*Previous ... Next*: Smith Number*Previous ... Next*: Square Number*Previous ... Next*: Powerful Number*Previous ... Next*: Square of Small-Digit Palindromic Number is Palindromic*Previous ... Next*: Semiprime Number

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $121$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $121$